uestion 1 (1 point) You invest \( \$ 10000.00 \) in a savings account that pays interest of \( 8 \% \) compounded monthly. What is the value of your account after 14 montl

To find the value of the account after 14 months with monthly compounding interest, we can use the formula for compound interest: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount (the initial amount of money). - \( r \) is the annual interest rate (decimal). - \( n \) is the number of times that interest is compounded per year. - \( t \) is the time the money is invested for in years. ### Given: - \( P = 10000 \) - \( r = 0.08 \) (8% as a decimal) - \( n = 12 \) (monthly compounding) - \( t = \frac{14}{12} \) years (since we need to convert months to years) Now, let's calculate \( A \) step by step. 1. Calculate \( \frac{r}{n} \): \[ \frac{r}{n} = \frac{0.08}{12} \] 2. Calculate \( nt \): \[ nt = 12 \times \frac{14}{12} = 14 \] 3. Substitute these values into the compound interest formula: \[ A = 10000 \left(1 + \frac{0.08}{12}\right)^{14} \] Now, let's perform the calculations. Calculate the value by following steps: - step0: Calculate: \(\frac{0.08}{12}\) - step1: Convert the expressions: \(\frac{\frac{2}{25}}{12}\) - step2: Multiply by the reciprocal: \(\frac{2}{25}\times \frac{1}{12}\) - step3: Reduce the numbers: \(\frac{1}{25}\times \frac{1}{6}\) - step4: Multiply the fractions: \(\frac{1}{25\times 6}\) - step5: Multiply: \(\frac{1}{150}\) Calculate or simplify the expression \( 12 * (14 / 12) \). Calculate the value by following steps: - step0: Calculate: \(12\left(\frac{14}{12}\right)\) - step1: Reduce the fraction: \(12\times \frac{7}{6}\) - step2: Reduce the numbers: \(2\times 7\) - step3: Multiply the numbers: \(14\) Calculate or simplify the expression \( 10000 * (1 + (0.08 / 12))^14 \). Calculate the value by following steps: - step0: Calculate: \(10000\left(1+\left(\frac{0.08}{12}\right)\right)^{14}\) - step1: Divide the terms: \(10000\left(1+\frac{1}{150}\right)^{14}\) - step2: Add the numbers: \(10000\left(\frac{151}{150}\right)^{14}\) - step3: Simplify: \(10000\times \frac{151^{14}}{150^{14}}\) - step4: Rewrite the expression: \(2500\times 4\times \frac{151^{14}}{150^{14}}\) - step5: Rewrite the expression: \(2500\times 4\times \frac{151^{14}}{50^{14}\times 3^{14}}\) - step6: Rewrite the expression: \(50^{2}\times 4\times \frac{151^{14}}{50^{14}\times 3^{14}}\) - step7: Reduce the numbers: \(4\times \frac{151^{14}}{50^{12}\times 3^{14}}\) - step8: Rewrite the expression: \(4\times \frac{151^{14}}{2^{12}\times 25^{12}\times 3^{14}}\) - step9: Rewrite the expression: \(2^{2}\times \frac{151^{14}}{2^{12}\times 25^{12}\times 3^{14}}\) - step10: Reduce the numbers: \(1\times \frac{151^{14}}{2^{10}\times 25^{12}\times 3^{14}}\) - step11: Multiply the fractions: \(\frac{151^{14}}{1024\times 25^{12}\times 3^{14}}\) Let's summarize the calculations step by step: 1. **Calculate \( \frac{r}{n} \)**: \[ \frac{0.08}{12} \approx 0.00666667 \] 2. **Calculate \( nt \)**: \[ nt = 14 \] 3. **Substituting into the compound interest formula**: \[ A = 10000 \left(1 + 0.00666667\right)^{14} \] Now, calculating \( A \): \[ A \approx 10000 \left(1.00666667\right)^{14} \] Using the calculated values: \[ A \approx 10000 \times 1.0996 \approx 10974.88 \] ### Final Result: The value of your account after 14 months is approximately \( \$ 10,974.88 \).